Compensated demand function

In microeconomics, a consumer's Hicksian demand correspondence is the demand of a consumer over a bundle of goods that minimizes their expenditure while delivering a fixed level of utility. If the correspondence is actually a function, it is referred to as the Hicksian demand function, or compensated demand function. The function is named after John Hicks.

Mathematically,

where h(p,u) is the Hicksian demand function, or commodity bundle demanded, at price level p and utility level ${\displaystyle {\bar {u}}}$. Here p is a vector of prices, and X is a vector of quantities demanded so that the sum of all p_ix_i, is the total expense on goods X.

Hicksian demand functions are often convenient for mathematical manipulation because they do not require income or wealth to be represented. Additionally, the function to be minimized is linear in the ${\displaystyle x_{i}}$, which gives a simpler optimization problem. However, Marshallian demand functions of the form ${\displaystyle x(p,w)}$ that describe demand given prices p and income ${\displaystyle w}$ are easier to observe directly. The two are trivially related by

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